Cubic Fields and Radical Extensions
Ming-chang Kang · American Mathematical Monthly · 2000
For example, if a is a root of X3 X 1 = 0, then Q(a) is not a radical extension of Q; on the other hand, if /8 is a root of X3 6x 6 = 0, then Q(,8) is a radical extension of G1I1. Note that 81v2 12u3 = 3A where A is the discriminant of the equation X3 ux v = 0. Similarly, u3/v2 is related to the characteristic-two discriminant; see Remark 4. Part of Theorem 1 may be found in [3], which contains an interesting result whose proof uses Galois theory: